Question: Simplify; express your answer in exponential form. Assume $z\neq 0, a\neq 0$. $\dfrac{{(z^{-4}a^{-1})^{-2}}}{{z^{-5}a^{2}}}$
To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(z^{-4}a^{-1})^{-2} = (z^{-4})^{-2}(a^{-1})^{-2}}$ On the left, we have ${z^{-4}}$ to the exponent ${-2}$ . Now ${-4 \times -2 = 8}$ , so ${(z^{-4})^{-2} = z^{8}}$ Apply the ideas above to simplify the equation. $\dfrac{{(z^{-4}a^{-1})^{-2}}}{{z^{-5}a^{2}}} = \dfrac{{z^{8}a^{2}}}{{z^{-5}a^{2}}}$ Break up the equation by variable and simplify. $\dfrac{{z^{8}a^{2}}}{{z^{-5}a^{2}}} = \dfrac{{z^{8}}}{{z^{-5}}} \cdot \dfrac{{a^{2}}}{{a^{2}}} = z^{{8} - {(-5)}} \cdot a^{{2} - {2}} = z^{13}$